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# The analytical solution for a stress field in an infinite plate with a circular inclusion due to an applied tensile stress

Tue, 2009-01-20 15:13 - Mike Tonks

I need to know the analytical solution for the stress field in an isotropic infinite plate with a circular inclusion, assuming linear elasticity. The plate has an applied tensile stress.

The solution for a hole in an infinite plate is very common (http://en.wikiversity.org/wiki/Introduction_to_Elasticity/Plate_with_hol...). I need a similar solution in which the inclusion has some elasticity tensor C and the matrix some tensor C_0 with C=x*C_0, where x is a scalar.

Thanks!

Mike

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## Eshelby Tensor

Mike,

Unfortunately I cannot give you any detailed information from where I am sitting right now, but the problem you describe has, to the best of my knowledge, been solved by Eshelby for circular and elliptical inclusions.

Start looking for "Eshelby tensor" and I am sure that you will get relevant hits!

I apologize again for not being more specific!

Best regards,

Thomas

## Semi-Infinite Thin Plate with a Circular Inclusion

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Shunsuke

Shioya “On a Semi-Infinite Thin Plate

with a Circular Inclusion under Uniform Tension” Japan Society of

Mechanical Engineers Vol. 10, No. 37, 1967.

you will find above papare online, hope that will help. However I am looking for the proper solution.

Regards

Gouse

## I am sorry

I am sorry in my above post there was some unwanted text, I don't know how it got added.

Shunsuke

Shioya “On a Semi-Infinite Thin Plate

with a Circular Inclusion under Uniform Tension” Japan Society of

Mechanical Engineers Vol. 10, No. 37, 1967.

you will find above paper online, hope that will help. However I am looking for the proper solution.

Regards

Gouse

## Thanks for your help

Gouse,

I found the paper and I think it might have what I need. If you do find the proper solution, please post it.

Thomas, thanks for your comment as well. I will look more into Eshelby's work.

Thanks!

Mike

## using complex variable method

Mike,

If you consider the analytical solution using complex variable method, I think text book by Muskhelishvili will help you a lot, "Some basic problems of the mathematical theory of elasticity", 1953, Groningen. Also another text book, "Complex Variable Method in Elasticity" by England.

Many papers have used that method for circular inclusion problems, whether under uniform tension (elasticity plane) or uniform heat flux (thermoelasticiy plane). Here is some example:

Fukui "The Elastic Plane with a Circular Insert, Loaded by a Concentrated Moment" JSME vol 10, no 37, 1967

Chao "Interaction between a crack and a circular elastic inclusion under remote uniform heat flow" Int J Solids Structures vol 33, no 26, 1996

I wish this comment appropriate with your need and will help you

Regards,

A Wikarta

## I found the solution

In the paper suggested by Gouse, I found a reference to a paper that had the complete solution. If anyone else might be interested in the solution, the paper is:

Report of the Aeronautical Research Institute, Tokyo Imperial University

Vol.6, No.68(19310400) pp. 25-43

http://ci.nii.ac.jp/naid/110004557431/

Thanks for your help,

Mike

## Inclusions and inhomogeneities-- analytical solution

Dear Mike,

I see that you have got the solution already. However, for the sake on completion, here is some more information that might be of use.

The method of complex variables for solving for the elliptic inclusion/inhomogeneity is also discussed in Jaswon and Bhargava's 1960 paper in Proceedings of Cambridge

Philosophical Society, (Volume 57, p. 669). Further, the monograph by Mura (Micromechanics of defects in solids) has these solutions as well as references to papers that give solutions to the same problem for materials other than isotropic (say, cubic, orthotropic and so on).

Guru